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(CNN) -- Hoping to speed up the identification of those killed by Hurricane Katrina and its aftermath, Louisiana said Thursday it would accept a new federal plan to pay for most of the cost for DNA testing.

The identification of bodies has been slowed in the state amid a bureaucratic fight over whether the state or federal government would foot the bill, estimated to be about $13 million.

Earlier this week, the Federal Emergency Management Agency informed state officials that funding the DNA testing was not its responsibility. The agency then backpedaled late Wednesday, saying it would pay the cost of the DNA testing through the Louisiana state police DNA laboratory.

Under that plan, FEMA will pay 100 percent of the cost through November 26. After that, the state would be left paying for about 10 percent of the cost, or roughly $1.3 million.

The proposal still needs to be approved by the Office of Management and Budget.

Denise Bottcher, a spokeswoman for Gov. Kathleen Blanco, said the governor hopes that FEMA will extend its offer and pick up the full tab, because the state is short of cash.

If FEMA declines, she said, the state would pay the 10 percent cost because it has a "moral obligation to do so."

The federal Department of Health and Human Services had offered to hire a contractor to do the DNA testing at no cost to the state.

But Bottcher said the governor was not interested in that option because it would take months to set up -- an unacceptable choice when families simply want to bury their loved ones.

Bottcher said testing DNA samples has been delayed because the police crime lab and FEMA have been negotiating over the price.

In all, 321 bodies remain unidentified in refrigerated trucks at a makeshift morgue set up after the hurricane. Of those, officials say about 150 will definitely need DNA testing for identification. Dental records and other means may be enough to identify the others.

"I'm just sick to my stomach," said Linda Hymel, whose younger brother, Darryl, is believed to be among those in the refrigerated trucks. "I'm not only grieving and mourning -- I'm angry. ... It's so simple: I just want my brother."

The bureaucratic delays have even frustrated the state's top coroner.

"I need DNA results," said Dr. Louis Cataldie. "I'm not the money man. I find myself in a position where money is holding me up right now."

In an interview with CNN Wednesday night, Blanco said the delay has been painful for families: "It is just part of the frustrating process that we have all been going through."

She added, "We feel very sorry for all these folks. It's been very complicated."